Linear Degenerate Tropical Flag Varieties

TL;DR

This paper explores linear degenerations of flag varieties through tropical geometry, establishing correspondences between various mathematical structures such as valuated matroids and tropical linear spaces.

Contribution

It introduces the linear degenerate flag Dressian and proves a new correspondence linking degenerations, valuated matroids, and tropical linear spaces.

Findings

Defined the linear degenerate flag Dressian.

Established a correspondence between points in the Dressian and degenerations.

Connected degenerations with morphisms of valuated matroids.

Abstract

We study linear degenerations of flag varieties from the point of view of tropical geometry. We define the linear degenerate flag Dressian and prove a correspondence between: $(a)$ points in the linear degenerate flag Dressian, $(b)$ linear degenerate flags of valuated matroids, $(c)$ linear degenerate flags of tropical linear spaces, and $(d)$ morphisms of valuated matroids arising from projection maps.